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When does strong duality fail in linear programming? I have considered the case when both primal and dual solutions are infeasible, but then there are no optimal solutions at all.

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The strong duality theorem of linear programming states that (quoted from "Algorithms" by S. Dasgupta, C. H. Papadimitriou, and U. V. Vazirani, 2006) "If a linear program has a bounded optimum, then so does its dual, and the two optimum values coincide." Hence, it never fails.

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There is a very good exercise - 5.23 - in Boyd's book on Convex Optimization (the solution might also help).

They prove that strong duality holds for the following LP and its dual provided at least one of the problems is feasible. In other words, the only possible exception to strong duality occurs when $p^*= \infty$ and $d^* = − \infty$.

LP:

$\min c^Tx$

st $Ax=b$

DUAL:

$max −b^Tz$

st $ A^T z + c = 0$ and $ z \succeq0$

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